EPJ Web of Conferences 66, 03014 (2014)
DOI: 10.1051/epjconf/ 201 4 6603014
C Owned by the authors, published by EDP Sciences, 2014
The ratio method: a new way to look at halo nuclei
P. Capel1 , a , R. C. Johnson2 , b , and F. M. Nunes3 , c
1
Physique Nucléaire et Physique Quantique, Université Libre de Bruxelles (ULB), B-1050 Brussels, Belgium
Department of Physics, University of Surrey, Guildford GU2 7XH, UK
3
National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA
2
Abstract. A new reaction observable is presented to study exotic loosely-bound structures, such as halo nuclei. It consists of the ratio of two angular distributions, e. g. one
for breakup and one for elastic scattering. This ratio is nearly independent of the reaction
mechanism and is very sensitive to the projectile structure. This new ratio method is
illustrated on the particular case of 11 Be, the archetypal one-neutron halo nucleus.
1 Introduction
The development of radioactive-ion beams (RIB) in the mid-80s, has enabled us to explore the nuclear
landscape away from stability. Various exotic structures have been uncovered thanks to this technological breakthrough, among which halo nuclei [1]. These light neutron-rich nuclei exhibit a much
larger matter radius than their isobars. This surprising property is understood as resulting from their
very low binding energy for one or two neutrons. Thanks to this lose binding, the valence neutrons
exhibit a high probability of presence at large distance from the other nucleons. They hence form
a sort of halo around a compact core that contains the other nucleons. The best known halo nuclei
are 11 Be and 15 C with a one-neutron halo, and 6 He and 11 Li with two neutrons in their halo. Albeit
possible, proton halos are less probable due to the presence of the Coulomb barrier between the core
and the valence protons, which hinder the development of a long tail in the wave function.
Being located on the shoulder of the valley of stability, halo nuclei cannot be studied through
usual spectroscopic techniques and one must resort to indirect methods to infer information about
their structure. Reactions, such as breakup or elastic scattering are among the most used tools to
study halo nuclei [2]. However, to extract valuable information from such experimental data, a good
understanding of the reaction process is required. Various reaction models have been developed with
this aim, e. g., the Continuum Discretized Coupled Channel method (CDCC), the time-dependent
model (TD), and the Dynamical Eikonal Approximation (DEA) (see Ref. [3] for a recent review and
Ref. [4] for a comparison of these three models). Theoretical analyses of reactions have shown that
higher-order effects such as coupling within the continuum lead to complicated reaction mechanisms
that hinder the analysis of reaction cross sections simply in terms of the projectile structure. Moreover,
a e-mail: [email protected]
b e-mail: [email protected]
c e-mail: [email protected]
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EPJ Web of Conferences
even state-of-the-art reaction models are sensitive to inputs such as the optical potentials used to
simulate the scattering of projectile components by the target. The core-target interaction in particular
is often unknown as the core is usually itself radioactive.
To circumvent this problem, we suggest a new observable that is nearly independent of the reaction
mechanism and that is therefore more sensitive to the projectile structure than usual breakup or elasticscattering cross sections [5, 6]. This observable consists of the ratio of angular distributions for two
different processes such as breakup and elastic scattering. It can be directly compared to the halo form
factor predicted by an adiabatic description of the reaction [7].
After describing the theoretical framework in which reactions are modeled, we introduce this new
observable. It is then tested using a fully dynamical model of reactions for the case of 11 Be impinging
on Pb at 69AMeV and C at 67AMeV [2]. We end this contribution by a brief conclusion and the
prospects of this work.
2 Theoretical framework
We consider reactions in which a one-neutron halo nucleus is impinging on a target T . The projectile
is modeled as a valence neutron n loosely bound to a core c. It is described by the Hamiltonian
H0 = T r + Vcn (r),
(1)
where r is the neutron-core relative coordinate, T r is the c-n kinetic-energy operator, and Vcn is a
potential that simulates the interaction between the halo neutron and the core. The parameters of
that potential are adjusted to reproduce the binding energy of the projectile and some of its low-lying
levels. The target T is supposed structureless and its interaction with the projectile components is
simulated by the optical potentials VcT and VnT . Within this framework, the theoretical study of
reactions involving halo nuclei reduces to solving the three-body Schrödinger equation
[T R + H0 + VcT (RcT ) + VnT (RnT )] Ψ(r, R) = ET Ψ(r, R),
(2)
where R is the projectile-target relative coordinate and T R is the corresponding kinetic-energy operator. The projectile is initially in its ground state ϕ0 while impinging on the target:
Ψ(r, R) −→ eiKZ+... ϕ0 (r),
(3)
Z→−∞
where K is the wave number of the initial relative motion of the projectile to the target, in direction b
Z.
3 The ratio idea
It has been observed that angular distributions for elastic scattering and breakup exhibit very similar
features: Coulomb rainbow, oscillatory patterns etc. [8]. This result can be qualitatively understood
within the Recoil Excitation and Breakup model (REB) [7], which relies on two simplifying assumptions: First it includes an adiabatic treatment of the excitation of the projectile. Second, it neglects
VnT . Under these two assumptions the three-body problem (2) can be solved semi-analytically and
the elastic-scattering cross section can be elegantly factorized (see Ref. [7] for details):
(
)
dσel
2 dσ
= |F0,0 |
.
(4)
dΩ
dΩ pt
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In that expression, the cross section for the elastic scattering for a pointlike projectile (dσ/dΩ)pt is
multiplied by a form factor that accounts for the extension of the halo
∫
F0,0 =
|ϕ0 (r)|2 eiQ· r dr,
(5)
where Q is proportional to the transferred momentum. A similar result can be obtained for breakup
(
)
dσbu
2 dσ
= |F E,0 |
,
(6)
dEdΩ
dΩ pt
where the form factor reads
2
∑ ∫
2
i
Q
·
r
ϕl jm (E, r)ϕ0 (r)e
|F E,0 | =
dr .
(7)
l jm
Besides the wave function of the initial bound state of the projectile ϕ0 , it includes ϕl jm (E), the wave
functions describing the c − n relative motion at energy E in the continuum within partial wave l jm.
The expressions (4) and (6) explain qualitatively why similar cross sections are observed for
breakup and elastic scattering [8]. Indeed both factorizations include the same point-like cross section, which describes most of the angular dependence of both cross sections. In addition, this analysis
lead to the idea of the ratio method. Within the REB the ratio of Eqs. (4) and (6) is a mere ratio of two
form factors and should be completely independent of the reaction mechanism. It should specially not
depend on the VcT interaction that is the most uncertain optical potential in the reaction model.
4
11
Be as a test case
In order to test the ratio method, we confront the qualitative REB predictions with calculations of a
fully dynamical reaction model, the DEA [9, 10], which does not rely on the adiabatic approximation,
and which includes the interaction between the halo neutron and the target VnT . As a test case, we
consider the collision of 11 Be on Pb at 69AMeV, which correspond to the RIKEN experiment [2],
with which the DEA is in excellent agreement [10]. Various tests have led us to choose instead of
the elastic-scattering cross section (4), the summed cross section dσsum /dΩ to evaluate the ratio [6].
This cross section corresponds to all quasi-elastic processes: elastic and inelastic scattering as well as
breakup to all energies E in the c-n continuum. Using this cross section the REB prediction for the
ratio becomes simply the form factor |F E,0 |2 (5) [5, 6]
dσbu /dEdΩ (REB)
= |F E,0 |2 .
dσsum /dΩ
(8)
The corresponding results are displayed in Fig. 1(left), where the breakup cross section (green dashed
lines) and the summed cross section (ratio to Rutherford in blue dotted line) can be compared to each
other. In agreement with the results of Ref. [8], they exhibit very similar patterns: Coulomb rainbow,
oscillations at large angles. . . Their ratio [(8), red solid line] is hence very smooth, indicating that
most of the sensitivity to the reaction mechanism is removed within this observable. This is further
confirmed by the fact that the dynamical ratio is nearly superimposed onto its REB prediction (7)
(thick grey line). Further tests have confirmed its independence to the reaction mechanism: the ratio
computed for a carbon target is nearly identical to the one obtained on lead (see Fig. 1(right)) [5, 6].
Moreover, we have observed that the ratio can give access simultaneously to both the binding energy
of the system and the orbital angular momentum of the halo neutron [5, 6]. It is also sensitive to the
radial wave function of the halo [5, 6]. Valuable structure information can thus be extracted from this
new observable.
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103
dσbu /dEdΩ
dσsum /dσR
Ratio
|FE,0 |2
2
10
10
1
0.1
10−2
10
−3
10−4
0
2
4
6
θ (deg)
8
10
(dσbu /dEdΩ)/(dσsum /dΩ) (MeV−1 )
EPJ Web of Conferences
0.1
10−2
10−3
|FE,0 |2
Be+Pb @ 69AMeV
11
Be+C @ 67AMeV
11
10−4
0
0.05
0.1
0.15
Q (fm−1 )
0.2
0.25
0.3
Figure 1. Left: ratio method illustrated on 11 Be impinging on Pb at 69AMeV. The breakup angular distribution
and the summed cross section are compared to each other. They exhibit very similar features, which vanish when
taking their ratio. This ratio is in excellent agreement with the form factor predicted by the REB model.
Right: comparison of the ratio obtained on Pb and C showing its independence on the reaction process [5, 6].
5 Conclusion
Halo nuclei have been discovered thanks to the development of RIBs [1]. Their very exotic structure has been the subject of many experimental and theoretical studies. It is studied mostly through
reactions: elastic scattering, breakup . . . Unfortunately the analysis of these reactions is hindered by
dynamical effects and uncertainties in the reaction-model inputs. The ratio method suggests a new
observable that is nearly independent of reaction processes and hence is less prone to model uncertainties. It consists of the ratio of angular distributions that is predicted by the REB to depend only on
halo form factors [5, 6]. This prediction is analyzed within the state-of-the-art reaction model DEA
for 11 Be impinging on Pb at 69AMeV and on C at 67AMeV. It is shown to be nearly independent of
the reaction mechanism and to be greatly sensitive to the projectile structure. In particular, both the
halo binding to the core and its orbital can be precisely determined using the ratio [6]. Further tests
should tell us whether it can be extended to other loosely-bound projectiles such as proton halos or
two-neutron halo nuclei.
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